1. ## Area

Let R be the region bounded by the graph of y=(1/x)(ln(x)), the x axis, and the line x=e. Find the area of the region.

So i found that (1/x)(lnx) intersects the x axis at 1 so to solve for the area we would find the intergral of (1/x)(lnx) from 1-e. How would i find the antiderivative of (1/x)(lnx) and is my method right? This is a Calc AB problem

2. Originally Posted by calc_help123
Let R be the region bounded by the graph of y=(1/x)(ln(x)), the x axis, and the line x=e. Find the area of the region.

So i found that (1/x)(lnx) intersects the x axis at 1 so to solve for the area we would find the intergral of (1/x)(lnx) from 1-e. How would i find the antiderivative of (1/x)(lnx) and is my method right? This is a Calc AB problem
Integrate the function from 1 to e.

use the substitution $\displaystyle \ln x = u$

$\displaystyle \frac{1}{x}dx = du$

The integral comes out $\displaystyle \frac{(\ln x)^2}{2}$

Finish it.

See the attached graph

3. I misunderstood this problem. when they said $\displaystyle x=e$ for some reason i thought they meant $\displaystyle x=e^y$ so then i tried to solve for y. now it makes sense.

so the final product would be $\displaystyle \int_1^e$$\displaystyle \frac{1}{x}\ln(x)$ = $\displaystyle .5$

correct?